{"id":"70d986ca-49d0-46fb-80ef-52b8b75833d6","arxiv_id":"2505.22965","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"The authors derive Fisher information matrices and Cramér-Rao bounds from normalized STM local density of states, giving disorder-sensitive lower bounds on electron energy and position variances.","lead":"The paper turns STM maps of the local density of states into information-geometric objects: Fisher information matrices and Cramér-Rao bounds that quantify disorder. The proposal gives a new, widely applicable data-analysis framework for disordered metals and topological insulators.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The universal applicability of the CRBs to STM hinges on Eq. (10) canceling an energy-independent tunneling prefactor and on a single-particle Born-interpretation of LDOS in Eq. (5); if either fails, the Fisher information bounds a different PDF than the electron probability density.","rationale":"The reader's weakest assumption is exactly the energy-independent tunneling prefactor and the single-particle Born-interpretation of the LDOS contained in Eqs. (4), (5), and (10). My reading of the manuscript confirms that the mathematical core is sound and self-contained: the Fisher information matrices are constructed correctly, the Cramer-Rao bounds follow from Cauchy-Schwarz, and the numerical checks are consistent. The only place where the central claim can fail is the mapping from measured conductance to the electron probability density. If the prefactor is energy-dependent, or if interactions make the LDOS a many-body spectral function, the extracted rho(x,E) is no longer the Born probability, and the bounds in Eqs. (20) and (37) constrain a different distribution. This is not an internal inconsistency, so I would not reject or change the reader's verdict; it is a scope issue that can be resolved by restricting the claims to non-interacting single-particle systems and to STM setups with a well-characterized, energy-independent tip, or by deriving the corrected bound for energy-dependent prefactors.","tokens_in":10887,"tokens_out":17434,"duration_ms":201000,"concrete_test":"Numeric test of Eq. (10) with an energy-dependent prefactor: in a 2D disordered tight-binding metal with exact eigenstates, compute the true rho(x,E) from Eq. (8) and the true electron energy variance. Simulate dI/dV = g(E) rho~(x,E) with g(E) = 1 + lambda (E/t)^2 for lambda = 0.1 and 1, apply Eq. (10) to extract rho_exp(x,E), and compute the FIM and CRB from both rho and rho_exp. If the CRB from rho_exp fails to bound the true variance (or deviates by more than 10% for a plausible lambda), the extraction protocol is not robust. Second check: diagonalize a 2-site Hubbard model at U/t = 4, compute the one-electron removal spectral function, normalize as in Eq. (10), and compare with the non-interacting Born form Sigma_n |psi_n(x)|^2 delta(E-E_n); a discrepancy would confirm that interactions require an explicit caveat.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The inequalities in Eqs. (20) and (37) are distribution-free: they hold by Cauchy-Schwarz for any positive normalized probability density. The load-bearing step is identifying the normalized STM conductance with the electron probability density rho(x,E). Equation (10) divides the conductance by its energy integral, but this cancels the prefactor in Eq. (4) only if |M|^2 and the tip DOS are energy-independent. Real STM tips have energy-dependent DOS and matrix elements; then the extracted rho_exp(x,E) = g(E) rho(x,E) / integral g(E') rho(x,E') dE' is a different PDF, and the CRB computed from it need not bound the true electron energy or position variance. Independently, Eq. (5) expresses the LDOS as a sum of |wavefunction|^2, which is a single-particle Born probability. For a correlated material, STM measures the single-particle spectral function, a many-body object, not a position-energy probability density for an electron. Thus the conclusion that the formalism applies to 'any kind of material' and 'any type of long or short range defects' overreaches beyond the non-interacting single-particle demonstration. This does not invalidate the internal derivation, but it is the critical bridge from the mathematics to the 'probing by STM' claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a new information-geometric analysis of STM-measured local density of states (LDOS). It normalizes the LDOS to form a position-conditional energy probability density and an energy-conditional position probability mass function, then defines a real-space Fisher information matrix (FIM) and an energy-space FIM. From these it derives Cramér-Rao bounds that lower-bound the local energy variance (Eq. (20)) and the spatial position variance (Eq. (37)). The formalism is demonstrated numerically on disordered 2D tight-binding models of a metal and of a Chern insulator in both topological phases.","tokens_in":11149,"tokens_out":7449,"duration_ms":77895,"significance":"The mathematical core of the paper is sound and self-contained: the derivations of Eqs. (15)-(20) and (33)-(36) are correct, no parameters are fitted, and the numerical positivity checks are consistent with the inequalities. The distribution-free character of the Cramér-Rao bounds is a genuine strength, and the idea of using the FIM as a disorder-induced metric on real space and energy space is novel for STM analysis. The main weakness is the bridge between measured tunneling conductance and an electron probability density: the extraction protocol in Eq. (10) requires energy-independent tunneling prefactors, and the Born-probability interpretation in Eq. (5) is a single-particle statement. Because the paper claims applicability to any material and any disorder, this bridge is load-bearing and needs to be either justified or explicitly qualified.","major_comments":[{"comment":"The experimental extraction protocol divides the conductance by its energy integral. This cancels the prefactor in Eq. (4) only under the stated assumption that |M|^2 and the tip DOS are energy-independent. For real STM tips with energy-dependent DOS or matrix elements, the extracted object is rho_exp(x,E) = g(E) rho(x,E) / integral g(E') rho(x,E') dE', which is a different probability density; the Cramér-Rao bound computed from it need not bound the true electron energy or position variance. Please either justify the energy-independence assumption for a concrete experimental setup, or restrict the claims and discuss the resulting systematic error.","section":"II A, Eqs. (4) and (10)"},{"comment":"The Born-probability interpretation is introduced via single-particle eigenstates |a_n(x)|^2 in Eq. (5). For interacting or correlated materials, STM measures the single-particle spectral function, a many-body object, which is not a joint position-energy probability density for an electron. The statements in the Introduction and Conclusions that the formalism applies to 'any kind of material' and 'any types of long or short range defects' therefore overreach beyond the non-interacting tight-binding demonstration in Sections II C and III C. Please narrow the universality claims or provide an explicit argument for how the formalism extends to interacting systems.","section":"I, Eq. (5), and Conclusions"}],"minor_comments":[{"comment":"The abstract contains a typo, 'bonuds' should be 'bounds'.","section":"Abstract"},{"comment":"For a discrete spectrum, rho(x,E) is a sum of delta functions, so the integral in Eq. (11) is formal unless a broadening is specified. The Lorentzian regularization is introduced only for the energy-space FIM in Eq. (32); please state explicitly how the real-space FIM is regularized in the numerics or define it directly through the fidelity expansion in Eq. (12) with broadened LDOS.","section":"II A, Eq. (11)"},{"comment":"The quantity described as 'the CRB' is actually the difference between the energy variance and the Cramér-Rao bound, Var(E) - d<I>^T I^{-1} d<I>. Consider renaming it 'CRB gap' or 'excess variance' to avoid confusion.","section":"Fig. 2 caption"},{"comment":"For consistency with Eq. (28), the PMF should be expressed in terms of |a_n(x)|^2 as defined in Eq. (7), including the sums over sublattice and orbital/spin indices, rather than |a_{n gamma}(x)|^2.","section":"III A, Eq. (31)"},{"comment":"The symbol f is used both for the arbitrary energy function in Eq. (14) and for the Fermi function in Eq. (24); please rename one of them to avoid ambiguity.","section":"II B, Eq. (24)"},{"comment":"The phrase 'under some unitary transformation U that changes the coordinates' is misleading because a unitary transformation in Hilbert space does not change spatial coordinates. This should be rephrased as a coordinate reparametrization or a rotation of the lattice axes.","section":"II B, text below Eq. (20)"}],"recommendation":"major_revision","confidential_remarks":"The central derivation is correct and the numerical checks are consistent, so the paper is not fatally flawed. The main issue is an overreach in the universality claims, which is fixable by rewriting the claims and adding a careful discussion of the experimental and many-body assumptions. The self-citations (Refs. 27 and 38) are used for motivation and context and are not load-bearing for the derivation. The paper fits the scope of cond-mat.str-el."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague—\n\nThe thing to know: this is a correct, clean application of textbook Fisher information and the Cramér-Rao bound to a new object, the normalized STM LDOS. The genuinely new step is treating rho(x,E) as a conditional probability density and building two Fisher information matrices, one in real space and one in energy space. The resulting inequalities, lower bounds on local energy variance and on position variance, could become useful quantitative disorder diagnostics for STM users. The derivations are self-contained, and the numerical positivity checks in Figs. 2 and 3 are consistent with the theory.\n\nThe paper deserves credit for honesty about the main modeling assumption. Equation (4) states the tunneling matrix element and tip DOS are energy-independent, and the extraction protocol in Eq. (10) divides that prefactor out. Within a single-particle lattice model the entire chain is sound. The authors also correctly note the real-space CRB is ill-defined where the FIM vanishes, e.g., homogeneous regions.\n\nThe soft spots are all about scope, not the internal math. The abstract and conclusion claim the bounds apply to 'any kind of material' and 'any type of long or short range defects.' That overreaches. If the STM prefactor depends on energy over the measured window, the extracted rho is a re-weighted PDF, and the CRB then bounds a different object, not the electron energy distribution. In a correlated material, STM measures the single-particle spectral function, which is not a Born position-energy probability. So the universality language should be dialed back to effectively single-particle systems with an explicit caveat about energy-dependent tip matrix elements. These are fixable in revision. Minor points: only one impurity realization per model, no error bars, no code or data shipped—acceptable for a proposal, but it tempers the word 'demonstrated.'\n\nThis paper is for people working on disordered electrons and STM data analysis. It deserves a serious referee, but with the expectation that the authors narrow the claims and state experimental assumptions more carefully.","headline":"Correct, clean application of standard Fisher-information math to a genuinely new object—normalized STM LDOS—but the universal-applicability claims outrun the demonstrated single-particle, energy-independent-tip regime.","tokens_in":11692,"tokens_out":3315,"would_cite":false,"duration_ms":35653,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that the normalized local density of states measured by STM is a probability density, and that its Fisher information matrices impose Cramér-Rao lower bounds on the local energy variance and the spatial variance of…","keywords":["Fisher information matrix","Cramér-Rao bound","local density of states","scanning tunneling microscopy","disorder","information geometry","Chern insulator","tight-binding model"],"falsifier":"Modify the same tight-binding models by replacing the constant tunneling prefactor in Eq. (4) with an energy-dependent $|M(E)|^2\\rho_{\\mathrm{tip}}(0,E)$ and recompute the ratio in Eq. (10); the resulting object is no longer the electron probability density, and the Fisher information and Cramér-Rao bounds computed from the normalized conductance would be bounds on a different quantity, which a direct comparison with the true quantum variances would expose.","tokens_in":10648,"feed_emoji":"🔬","tokens_out":9326,"duration_ms":90827,"temperature":0.7,"pith_summary":"The paper treats the normalized local density of states $\\rho(x,E)$ measured by STM as a conditional probability density: the probability that an electron at position $x$ has energy $E$ within the probe window. From this single object it builds two Fisher information matrices—one in real space and one in energy space—and shows that the corresponding Cramér-Rao bounds constrain the variance of local energy and the variance of electron position. Because the bounds are derived for any normalized probability density, the authors argue they hold for any host material and any type of disorder, and they demonstrate the quantities on disordered tight-binding models of a metal and a Chern insulator. The payoff is a new, model-free way to interpret STM conductance maps as information geometry of the disordered solid.","feed_headline":"STM maps yield lower bounds on electron energy and position variance","feed_subtitle":"Treating STM conductance as a probability density puts hard limits on local energy and position variance.","key_machinery":"The load-bearing object is the normalized local density of states $\\rho(x,E)=\\tilde{\\rho}(x,E)/\\int_{E_c}^{0}dE\\,\\tilde{\\rho}(x,E)$, a conditional probability density whose variation defines the two Fisher informations: $I_{\\mu\\nu}(x,E_c)=\\int dE\\,\\partial_\\mu\\rho\\,\\partial_\\nu\\rho/\\rho$ in real space and $I_E(E)=\\sum_x (\\partial_E\\rho)^2/\\rho$ in energy space. The bounds are carried by the positive semidefiniteness of the correlation matrix built from $(f-\\langle f\\rangle,\\partial_\\mu\\ln\\rho)$, and by the Cauchy-Schwarz inequality in the energy-space case, which converts the overlap of normalized LDOS at nearby positions or energies into an exact variance bound. Numerically, spatial derivatives are evaluated by central differences along lattice vectors, and the energy derivatives of the delta functions entering $\\rho$ are computed with a Lorentzian broadening.","core_discovery":"The central claim is that the STM differential conductance, after summing over sublattices and normalizing by its energy integral, is exactly the electron probability density $\\rho(x,E)$ of an electron at position $x$ with energy $E$, provided the tunneling matrix element and tip density of states are energy-independent. Treating $E$ as the random variable and $x$ as the parameter yields a real-space Fisher information matrix $I_{\\mu\\nu}(x,E_c)$, whose Cramér-Rao bound reads $\\mathrm{Var}[f] \\ge \\partial_\\mu\\langle f\\rangle I^{\\mu\\nu}\\partial_\\nu\\langle f\\rangle$; choosing $f(E)=E$ bounds the local energy variance at each site. Treating $x$ as the random variable and $E$ as the parameter yields an energy-space Fisher information $I_E(E)$, whose bound $\\mathrm{Var}[x_\\mu] \\ge (\\partial_E\\langle x_\\mu\\rangle)^2/I_E$ bounds the spatial variance at fixed energy. The paper verifies numerically in 2D metals and in Chern insulators in both topological phases that these quantities are positive, that disorder induces nonzero Fisher information and a curved information geometry, and that the bounds are respected at every site and every energy.","pith_inferences":["One testable extension is to use the real-space Fisher information texture as a model-free fingerprint of the impurity configuration: since $I_{\\mu\\nu}$ is largest where $\\rho$ changes sharply, STM maps could in principle be inverted to locate or classify disorder without assuming a specific Hamiltonian.","The same probability-density construction could be carried over to other scanning spectroscopies that produce normalized maps over a coordinate and a bias parameter, whenever a legitimate probability interpretation exists; this is an extrapolation beyond the paper's solid-state examples.","The larger energy variances and Cramér-Rao values seen in the topologically nontrivial Chern phase could in principle serve as a disorder-sensitive indicator of topology, but the paper does not claim such a diagnostic; a systematic scan over disorder strength would be needed to test it."],"forward_implications":["Any STM dataset that resolves $dI/dV(x,E)$ over a grid can be processed through the ratio in Eq. (10) to produce maps of the real-space Fisher information and its volume form, making impurity-induced curvature of space a directly measurable field.","Wherever the Fisher information matrix is invertible, the local energy variance is bounded below by the gradient of the local energy via $\\mathrm{Var}[E] \\ge \\partial_\\mu\\langle E\\rangle I^{\\mu\\nu}\\partial_\\nu\\langle E\\rangle$, a bound that matters only where disorder makes the Fisher information nonzero.","At fixed energy, the spatial variance of electrons is bounded below by $(\\partial_E\\langle x_\\mu\\rangle)^2/I_E$, which is nonzero precisely in disordered systems, so disorder strictly raises the lower bound on position fluctuations compared with a homogeneous system.","Because the derivation uses only that $\\rho$ is a normalized probability density, the two Cramér-Rao bounds apply equally to metals, insulators, semiconductors, and topological phases, as illustrated by the Chern insulator results."],"supporting_citations":[{"why":"Supplies the phenomenological STM conductance-to-LDOS relation used in Eq. (4), from which the normalized probability density is extracted.","marker":"[12]"},{"why":"Defines the Fisher information that the paper adapts to build the real-space and energy-space FIMs from the normalized LDOS.","marker":"[13]"},{"why":"Provides the spatially resolved quantum metric construction and the central-difference derivative used for the numerical real-space FIM.","marker":"[27]"},{"why":"Provides the fidelity expansion between nearby states that motivates interpreting the FIM as a metric, as in Eq. (12).","marker":"[28]"},{"why":"Gives the variance-bound derivation the paper adapts to obtain the Cramér-Rao bounds on energy and position variances.","marker":"[29]"},{"why":"Defines the Chern insulator lattice model used to demonstrate the bounds in a topological phase.","marker":"[31]"}],"fun_headline_variants":["STM maps place bounds on electron energy and position variance","Fisher information from STM sets limits on electron localization","Cramér-Rao bounds from STM data constrain electron spread","Disorder in STM yields new limits on electron variance","STM reveals geometric bounds on electron energy and position"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the STM differential conductance is proportional to the single-particle local density of states with an energy-independent tunneling matrix element and tip density of states, so the normalized conductance equals the electron's position-energy probability density.","fun_headline_variants_meta":{"raw":{"variants":["STM maps place bounds on electron energy and position variance","Fisher information from STM sets limits on electron localization","Cramér-Rao bounds from STM data constrain electron spread","Disorder in STM yields new limits on electron variance","STM reveals geometric bounds on electron energy and position"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000211,"raw_usage":{"total_tokens":1415,"prompt_tokens":946,"completion_tokens":469,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":562,"completion_tokens_details":{"reasoning_tokens":390}},"tokens_in":562,"tokens_out":469,"duration_ms":5525,"temperature":1.0,"reasoning_tokens":390,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:56:45.060617+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Modify the same tight-binding models by replacing the constant tunneling prefactor in Eq. (4) with an energy-dependent $|M(E)|^2\\rho_{\\mathrm{tip}}(0,E)$ and recompute the ratio in Eq. (10); the resulting object is no longer the electron probability density, and the Fisher information and Cramér-Rao bounds computed from the normalized conductance would be bounds on a different quantity, which a direct comparison with the true quantum variances would expose.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the phenomenological STM conductance-to-LDOS relation used in Eq. (4), from which the normalized probability density is extracted."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the spatially resolved quantum metric construction and the central-difference derivative used for the numerical real-space FIM."},{"cited_title":"van den Bos ,\\ @noop title Parameter Estimation for Scientists and Engineers \\ ( publisher John Wiley and Sons, Hoboken ,\\ year 2007 ) NoStop","cited_arxiv_id":null,"evidence_quote":"Gives the variance-bound derivation the paper adapts to obtain the Cramér-Rao bounds on energy and position variances."},{"cited_title":"Chen ,\\ 10.1103/PhysRevB.101.195120 journal journal Phys","cited_arxiv_id":null,"evidence_quote":"Defines the Chern insulator lattice model used to demonstrate the bounds in a topological phase."}],"review_version":1}